High-frequency data in finance have led to a deeper understanding on probability distributions of market prices. Several facts seem to be well established by empirical evidence. Specifically, probability distributions have the following properties: (i) They are not Gaussian and their center is...

We studied random walks on two-dimensional patterns formed by the sequence of configurations of complex elementary cellular automata (CA) with random initial configurations. The walkers are allowed to jump between nearest neighbours or next nearest neighbours 1 sites. On patterns of rules 22,...

The random walk simulation of a Levy flight shows a linear relation between the mean square displacement 〈r2〉 and time. We have analyzed different aspects of this linearity. It is shown that the restriction of jump length to a maximum value (lm) affects the diffusion coefficient, even though...

Stochastic variables whose addition leads to q-Gaussian distributions Gq(x)∝[1+(q-1)βx2]+1/(1-q) (with β0, 1⩽q3 and where [f(x)]+=max{f(x),0}) as limit law for a large number of terms are investigated. Random walk sequences related to this problem possess a simple additive–multiplicative...

As well known, the generalized Langevin equation with a memory kernel decreasing at large times as an inverse power law of time describes the motion of an anomalously diffusing particle. Here, we focus attention on some new aspects of the dynamics, successively considering the memory kernel, the...

A diffusion model based on a continuous time random walk scheme with a separable transition probability density is introduced. The probability density for long jumps is proportional to x−1−γ (a Lévy-like probability density). Even when the probability density for the walker position at...

We study the relationship between anomalous diffusion and persistent motion of micron-sized particles moving in a viscoelastic environment and subjected to an external noise. In the framework of a generalized Langevin equation, we compare the analytical expressions of the mean square...

In the case of time-fractional diffusion–wave equation considered in the spatial domain −∞x∞, evolution of initial box-signal was investigated by Mainardi [F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos Solitons Fractals 7 (1996)...

This work is devoted to investigating exact solutions of generalized nonlinear fractional diffusion equations with external force and absorption. We first investigate the nonlinear anomalous diffusion equations with one-fractional derivative and then multi-fractional ones. In both situations, we...

We analyse a N-dimensional anisotropic nonlinear Fokker–Planck equation by considering stationary and time-dependent solutions. The stationary solutions are obtained for very general situations, including those when the diffusion coefficients are spatial dependents. Time-dependent solutions...